Unit #4: Quadratic - Highs and Lows (13 days + 2 jazz + 3 midterm summative evaluation days)

Transcription

1 Unit #4: Quadratic - Highs and Lows (13 days + 2 jazz + 3 midterm summative evaluation days) BIG Ideas: Investigate the three forms of the quadratic function and the information that each form provides. Using technology, show that all three forms for a given quadratic function are equivalent. Convert from standard (expanded) form to vertex form by completing the square. Sketch the graph of a quadratic function by using a suitable strategy. (i.e. factoring, completing the square and applying transformations) Explore the development of the quadratic formula and connect the value of the discriminant to the number of roots. Collect data from primary and secondary sources that can be modelled as a quadratic function using a variety of tools. Solve problems arising from real world applications given the algebraic representation of the quadratic function. DAY Lesson Title & Description 2P 2D Expectations 1 Graphs of quadratics in factored Form The zeros and one other point are necessary to have a unique quadratic function Determine the coordinates of the vertex from your sketch or algebraic model C No a C QF2.09 sketch graphs of quadratic functions in the factored form f (x) = a(x r )(x s) by using the x- intercepts to determine the vertex; Teaching/Assessment Notes and Curriculum Sample Problems Computer and data projector (Optional) Lesson Included Investigating the roles of a, h and k in the Vertex From Investigate the roles of a, h and k Apply a series of transformation to y=x 2 to produce the necessary quadratic function 2 Lesson Included N C QF2.05 determine, through investigation using technology, and describe the roles of a, h, and k in quadratic functions of the form f (x) = a(x h) 2 Computer Lab + k in terms of transformations on the graph of f(x) = x 2 (i.e., translations; reflections in the x-axis; vertical stretches and compressions) Sample problem: Investigate the graph f (x) = 3(x h) for various values of h, using technology, and describe the effects of changing h in terms of a transformation. 1

2 3 Sketching quadratics functions in vertex form Apply a series of transformation to y=x 2 to produce the necessary quadratic function N C QF2.06 sketch graphs of g(x) = a(x h) 2 + k by applying one or more transformations to the graph of f (x) = x 2 Computer and data projector Sample problem: Transform the graph of f (x) = x 2 to sketch the graphs of g(x) = x 2 4 and h(x) = 2(x + 1) 2 Lesson Included 4 Changing from vertex form to standard (expanded) form Verify using technology that both forms are equivalent N N QF2.07 express the equation of a quadratic function in the standard form f(x) = ax 2 + bx + c, given the vertex form f(x) = a(x h) 2 + k, and verify, using graphing technology, that these forms are equivalent representations or Sample problem: Given the vertex form f(x)= 3(x 1) 2 + 4, express the equation in standard form. Use technology to compare the graphs of these two forms of the equation. 5, 6 Completing the Square Use algebra tiles to investigate procedures Verify using technology that both forms are equivalent Develop a procedure to complete the square using algebra N C QF2.08 express the equation of a quadratic function in the vertex form f (x) = a(x h) 2 + k, given the standard form f(x) = ax 2 + bx + c by completing the square (e.g., using algebra tiles or diagrams; algebraically), including cases where b a is a simple rational number (e.g., ½, 0.75), and verify, using graphing technology, that these forms are equivalent representations; Algebra Tiles Day 5 Day 6 Lessons Included 7 Gathering information from the three forms of quadratic functions Use inspection to gather information N N QF2.10 describe the information (e.g., maximum, intercepts) that can be obtained by inspecting the standard form f (x) = ax 2 + bx + c, the vertex form f (x) = a(x h) 2 + k, and the factored form f (x) = a(x r)(x s) of a quadratic function; 8 Sketching the graph of quadratic functions in standard form Use a suitable strategy to gather information to construct the graph N R QF2.11 sketch the graph of a quadratic function whose equation is given in the standard form f (x) = ax 2 + bx + c by using a suitable strategy (e.g., completing the square and finding the vertex; factoring, if possible, to locate the x-intercepts),and identify the key features of the graph (e.g., the vertex, the x- and y-intercepts, the equation of the axis of symmetry, the intervals where the function is positive or negative, the intervals where the function is increasing or decreasing). 2

3 9 CAS ing out the quadratic formula Explore the development of the quadratic formula using CAS Apply the formula to solve equations using technology Lesson Included N R QF1.06 explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development, with technology, such as computer algebra systems, or without technology [student reproduction of the development of the general case is not required]), and apply the formula to solve quadratic equations, using technology; CAS 10 Relating roots and zeros of quadratic functions X-intercepts (zeros) and roots are synonymous The sign of the disciminant determines the number of roots N N QF1.07 relate the real roots of a quadratic equation to the x- intercept(s) of the corresponding graph, and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if b 2 4ac < 0); 11 Solving quadratic equations Solve equations using a variety of strategies Describe advantages and disadvantages of each strategy N C QF1.08 determine the real roots of a variety of quadratic equations (e.g., 100x 2 = 115x + 35), and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula) Sample problem: Generate 10 quadratic equations by randomly selecting integer values for a, b, and c in ax 2 + bx + c = 0. Solve the equations using the quadratic formula. How many of the equations could you solve by factoring?). 3

4 12, , 16 Nano Project or Fuel Fit Collect data from primary or secondary sources without technology Determine the equation of a quadratic model for the collected data using technology Solve problems from real world applications given the algebraic representation of a quadratic function Lessons Included Instructional jazz day Note: This day may be located throughout the unit as needed. Midterm summative assessment performance task Note: Two possible performance tasks are included (A Leaky Problem or Bridging the Gap) 17 Unit Review 18 Pencil and paper summative assessment on expectations from this unit not covered in the summative performance task. C C QF3.01 N N QF3.02 C C QF3.03 collect data that can be modelled as a quadratic function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials; measurement tools such as measuring tapes, electronic probes, motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E- STAT), and graph the data determine, through investigation using a variety of strategies (e.g., applying properties of quadratic functions such as the x-intercepts and the vertex; using transformations), the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and compare this equation to the equation of a curve of best fit generated with technology (e.g., graphing software, graphing calculator); solve problems arising from real-world applications, given the algebraic representation of a quadratic function (e.g., given the equation of a quadratic function representing the height of a ball over elapsed time, answer questions that involve the maximum height of the ball, the length of time needed for the ball to touch the ground, and the time interval when the ball is higher than a given measurement) Sample problem: When a 3 x 3 x 3 cube made up of 1 x 1 x 1 cubes is dipped into red paint, 6 of the smaller cubes will have 1 face painted. Investigate the number of smaller cubes with 1 face painted as a function of the edge length of the larger cube, and graph the function.) Sample problem: When a 3 x 3 x 3 cube made up of 1 x 1 x 1 cubes is dipped into red paint, 6 of the smaller cubes will have 1 face painted. Investigate the number of smaller cubes with 1 face painted as a function of the edge length of the larger cube, and graph the function.) or (Day 16) Note: The two midterm summative performance tasks are in the file Midterm SP Task. 4

5 Unit 4 : Day 1: Graphs of Quadratics in Factored Form Grade 11 U/C Minds On: 30 Action: 30 Consolidate:15 Total=75 min Minds On Action! Description/Learning Goals Materials BLM Activate prior knowledge on the factored form of a quadratic function BLM Sketch graphs of quadratic functions in the factored form f(x) = a(x r)(x s) Chart paper Determine the coordinates of the vertex using the graph or algebraic model Markers Computer & data projector FRAME document Assessment Opportunities Small Groups Discussion/Exploration In groups of three or four, students will use the graphic organizer (BLM 4.1.1) to activate prior knowledge on the properties of parabolas, the information they need to graph a parabola, the information they can gather from a quadratic equation in factored form and the steps they would need to determine the coordinates of the vertex. Groups will share their results with the class. The class will summarize the results on chart paper using a graphic organizer on chart paper and this will then be posted in the room. Optional: Use the Geometer s Sketchpad prepared sketch with a data projector to demonstrate the information that can be obtained from a graph of an equation in factored form. Pairs Practice In pairs, students will complete the practice questions on BLM using their activated prior knowledge. Ensure groups have students coming from both 2D and 2P in them. Cooperative Learning Strategy To share results each group can provide one piece of information. Continue cycling through the groups. If a group has no new information to share they may pass. Geometer s Sketchpad: Barge.gsp Consolidate Debrief Whole Class Discussion Students share their graphs of quadratic equations from BLM Pose the following Guiding Questions: o Are the two zeros enough information to make a unique graph? o What other information do you need to make a graph unique? o How does the value of a affect the graph? Mathematical Process Focus: Reflecting ( In their journals students will reflect on the new concept and correct their partner s answers. Students should correct each other s work. Journal Home Activity or Further Classroom Consolidation Assign further practice questions as needed. Update FRAME graphic organizer document with information from today s lesson. Have students write in their journals using one or more of the following prompts:. o What patterns did you see when the value of a changed? o How can you determine from the factored form if the sketch will open up or down? o How can you determine the coordinates of the vertex given the factored form? 5

6 4.1.1 Graphs of Quadratic Functions in Factored Form: Properties In groups, use the graphic organizer to gather information for graphing quadratic functions in factored form. Properties: Information needed to graph: Rules/Methods: y = a(x r)(x s) Examples: 6

7 4.1.2 Graphs of Quadratic Functions in Factored Form: Practice With a partner, use the information from your graphic organizer to graph the following quadratic functions: y = (x + 3)(x + 5) X-intercepts: Vertex: y = 2(x + 1)(x 3) y = -3(x 3)(x + 1) X-intercepts: Vertex: X-intercepts: Vertex: y = ( x 2)( x 4) X-intercepts: Vertex: 7

8 Unit 4 : Day 2: Investigating the roles of a, h & k in the vertex form Grade 11 U/C Minds On: 20 Action: 40 Consolidate:15 Total=75 min Minds On Action! Consolidate Debrief Concept Practice Description/Learning Goals Determine, through investigation using technology, and describe the roles of a, h, and k in quadratic functions in vertex form. Apply a series of transformations to the graph of f(x) = x 2 to produce the necessary graph Whole Class Check for Understanding In groups of four or five, students use the Graffiti strategy to summarize the roles of a, h, and k in quadratic functions of the form f(x)= a(x-h) 2 + k. Teacher can briefly assess the students understanding. Pairs Investigate Using the ParabolaSlider pre-made Geometer s Sketchpad sketch, students will investigate the role of a, h, and k in the quadratic function of the form f(x) = a(x h) 2 + k and record their finding on BLM Students demonstrate their understanding by completing BLM Whole Class Summarizing Mathematical Process Focus: Reasoning and Proving (Students to share how they completed the BLM by recognizing the characteristics of each equation that give the values of a, h, and k, vertex, # of x-intercepts, domain and range). Discuss conclusions from the investigation and worksheet and have students summarize their results on BLM Ensure that students understand the terminology o Translations o Reflections in the x-axis o Vertical stretches or compression Home Activity or Further Classroom Consolidation Assign further practice questions as needed. Update FRAME graphic organizer document with information from today s lesson. Assessment Opportunities Materials BLM BLM BLM BLM Computer lab Data projector Optional Activity: Masking tape FRAME document Students should be made aware of the difference between parameters and variables Literacy strategy: Use Graffiti, (Think Literacy Cross- Curricular Approaches, Grades 7-12, p 26) After the Minds On, class should move to the computer lab. (ParabolaSlider.gsp) Optional: Teacher could have students kinaesthetically demonstrate the roles of a, h, and k. Use tape to make a set of axes on the floor tiles of the classroom. Place seven students on the axes in the shape of x 2. Have the students transform for various values of a, h and k. After each question the students return to x 2. Have the other students coach participating and students and.change student roles frequently. 8

9 4.2.1 Investigation What are the roles of a, h and k? With a partner and using the Parabola slider Geometer s Sketchpad sketch, investigate the roles of a, h and k in the vertex form of the quadratic function. Record your findings. Role of a: As I increase the value of a (larger than 1), I notice Role of h: As I increase the value of h, I notice As I decrease the value of a (smaller than -1), I notice As I decrease the value of h ( h becomes negative), I notice As I change the value of a between -1 and 1, I notice When the value of h is zero, I notice Role of k: As I increase the value of k, I notice y = a(x h) 2 + k Other Observations: As I decrease the value of k (k becomes negative), I notice When the value of k is zero, I notice 9

10 4.2.2 Demonstrating understanding of the roles of a, h & k in y = a(x h) 2 + k In pairs, complete the following table. Equation Value Value of a of h Value of k Vertex (h, k) # of x- intercepts Transformations Starting from y=x 2 Domain & Range y = 3(x - 2) a = 7 h = 2 k = 1 (2, 1) None Vertical stretch by a factor of 3 Translated 2 units right Translated 1 unit upwards D: Set of real numbers R: y 1 y = -2(x - 3) y = 2 1 (x +1) 2 +5 y = 0.3(x + 2) y = ( x 4) - 8 y = 2x y = -3(x + 5) 2 10

11 4.2.3 Summarizing the roles of a, h & k in y = a(x h) 2 + k In a class discussion complete the following graphic organizer to summarize the roles of a, h and k. Role of a: Direction of Opening: When a is positive, the parabola opens. When a is negative, the parabola opens. Shape: If a > 1 or a < -1, then the graph of y = a(x h) 2 + k has an opening than y = 1(x h) 2 + k. If a is between -1 and 1, then the graph of y = a(x h) 2 + k has an opening than y = 1(x h) 2 + k. Role of k: Properties: If k > 0, then the graph of y = a(x h) 2 + k is translated vertically k units. If k < 0, then the graph of y = a(x h) 2 + k is translated vertically K units. Relation to the Vertex: The value of k is the - coordinate of the vertex. y = a(x h) 2 + k Role of h: Properties: If h > 0, then the graph of y = a(x h) 2 + k is translated horizontally h units to the. If h < 0, then the graph of y = a(x h) 2 + k is translated horizontally h units to the. Relation to the Vertex: The value of h is the - coordinate of the vertex. Example: y = -2(x 3) State: a) Direction of opening: b) Stretch or Compression: c) Transformations: X-Intercepts: If k = 0, then the graph has zero (x-intercept). If k > 0, then the graph has zeros (x-intercepts). If k < 0, then the graph has zeros (x-intercepts). d) Coordinates of the vertex: e) Number of x-intercepts: f) Domain and Range: 11

12 4.2.4 Function Aerobics PowerPoint Presentation File (Teacher) (Function Aerobics.ppt) 12

13 4.2.4 Function Aerobics PowerPoint Presentation File (Teacher) (continued) 13

14 4.2.4 Function Aerobics PowerPoint Presentation File (Teacher) (continued) 14

15 4.2.4 Function Aerobics PowerPoint Presentation File (Teacher) (continued) 15

16 Unit 4 : Day 3 : Sketching Graphs of f(x) = a(x h) 2 + k Grade 11 U/C Minds On: 15 Action: 35 Consolidate:25 Total=75 min Minds On Description/Learning Goals Sketch the graphs of f(x) = a(x h) 2 + k by applying one or more transformations to the graph of f(x) = x 2. Whole Class Demonstration Using a computer with data projector, lead function aerobics using the presentation file. Students will participate in function aerobics to show their understanding of the roles of a, h, and k in quadratic functions of the form f(x) = a(x h) 2 + k in terms of transformations on the graph of f(x) = x 2 Assessment Opportunities Materials BLM BLM Computer & data projector Optional: (Music & CD Player) Scissors Overhead projector Acetate sheets Chart paper Markers Clear a large central space in the classroom to allow students to perform function aerobics. Functions Aerobics.ppt Action! Consolidate Debrief Pairs Explore Students will explore how to sketch the graphs from the given equations by applying a series of transformations. Students record the results on BLM Whole Class Check for Understanding Using an overhead copy of BLM have students demonstrate how they found the graphs. Encourage many pairs of students to participate. Mathematical Process Focus: Communication (Students will communicate their answers using suitable mathematical vocabulary and using various representations e.g. graphs and verbal descriptions.) Optional background music could be used. Literacy strategy: Think/Pair/Share should be used during the action portion of the lesson. Teachers should print BLM on acetate sheets and provide each pair with copies. Teacher should ensure that all students have an opportunity to use the manipulative in their respective groups. Teacher should ensure proper use of terminology in student responses. Concept Practice Home Activity or Further Classroom Consolidation Assign extra practice questions as needed. 16

17 4.3.1 Manipulatives for Investigating the Graphs of Quadratic Functions in Vertex Form (Teacher) Photocopy the following onto acetate sheets. Each pair is to receive a set of three parabolas. Note: The parabolas have the same scale as grids on BLM y = x 2 y = 2x 2 y = x y = x 2 y = 2x 2 y = x y = x 2 y = 2x 2 y = x y = x 2 y = 2x 2 y = x

18 4.3.2 Investigating the Graphs of Quadratic Functions in Vertex Form For each quadratic function below you and your partner will: a) Describe the transformations that have transformed y=x 2 into the given quadratic function. b) Choose the appropriate shaped parabola on acetate and place the parabola on the grid with the vertex at the origin and opening upwards. c) Transform the parabola according to your list of transformations. d) Record a sketch the final parabola on the grid. Quadratic Function y = x Transformation(s) No stretch or compression No horizontal translations Vertical translation of 5 units upwards Graph y = 2(x 3) y = 1 ( x + 6)

19 4.3.2 Investigating the Graphs of Quadratic Functions in Vertex Form (continued) Quadratic Function Transformation(s) Graph y = -2(x + 3) y = -(x + 3) 2-6 y = 1 (x + 5) y = -3(x 1)

20 Unit 4 : Day 5 : Completing the square using algebra tiles Grade 11 U/C Minds On: 15 Action: 40 Consolidate:20 Total=75 min Minds On Action! Description/Learning Goals Express the equation of a quadratic function in the vertex form f(x) = a(x h) 2 + k, given the standard from f(x) = ax 2 + bx + c by completing the square, including cases where b/a is a simple rational number, and verify, using graphing technology, that these forms are equivalent representations. Students will be able to change from the standard from of a quadratic function to the vertex form of the quadratic function using algebra tiles. Whole Class Discussion Discuss quadratic functions in vertex form; f(x) = a(x h) 2 + k and ask guiding questions: o What information can we gather from the equation? (i.e. vertex, direction of opening, stretches, compressions) o How is this information helpful to us? o How could we rewrite quadratic equations in standard from to vertex form? Pairs Exploration Students will use Algebra Tiles to complete BLM Here are some guiding questions for students to think about while they are working on BLM o Which one of the tiles represents a square all on its own? o Does (x+1) 2 = x 2 + 1? o Discuss why is x 2 + 2x + 1 a square? o What does the value of a mean in term of algebra tiles (Answer- Number of squares to build)? Assessment Opportunities Materials BLM Algebra tiles (one set per pair) This is day one of a two-day investigation. Students would benefit from a review on how to use algebra tiles, especially, what each piece represents, i.e. the dimensions and the area. Literacy strategy: A tells B Have partners explain to each other how they completed their square. Consolidate Debrief Mathematical Process Focus: Reasoning and Proving (Students use algebra tiles to complete the square, trying different strategies, looking for a patterns and use logical reasoning.) Whole Class Discussion Have students share their responses from question 2 on BLM Possible guiding questions: o What observations can we draw from your experience with the algebra tiles? o Discuss limitations of algebra tiles You cannot use algebra tiles to complete the square for expressions that have negative coefficients, expressions where b is odd, or expressions where a does not divide evenly into b. Reflection Home Activity or Further Classroom Consolidation Teacher to assign further questions as needed. Have students write in their journals. Using the following prompts: o Why is completing the square an appropriate name for the procedure of converting a quadratic function into vertex form? o What types of quadratic expressions cannot be completed using algebra tiles? The second journal prompt will be a transition to the algebraic procedure for completing the square in the next lesson. 20

21 4.5.1 Completing the Square Algebra Tile Investigation Recall the values of each of the algebra tiles. The value of the tile is its area. We will only be working with positive (red tiles) representations of algebraic expressions. x 2 Tile Area = x x = x 2 units x Tile Area = 1 x = x units Unit Tile Area = 1 1 = 1 unit Your Task: With a partner, complete the following investigation. 1. Complete the table on the next two pages using algebra tiles. a) Represent each expression using the appropriate number of each of the algebra tiles. b) Using the tiles you selected, try to create a square of tiles. When doing so, keep the following rules in mind: You may only use one x 2 -tile in each square. You must use all the x 2 and x-tiles. Unit tiles are the only ones that can be leftover or borrowed. If you need more unit tiles to create a square you have to borrow them. The number you borrow will be a negative quantity. You may create multiple squares, but they must have the same size 2. After you have completed the table, answer the following questions: a) What strategy did you use to place the x tiles around the x 2 tile? b) Can you create a square with an odd number of x tiles? c) What is the relationship between the value of a and the number of squares you created? d) What is the name of the form for the combined expression in the last column? e) Why is Completing the Square an appropriate name for this procedure? 21

22 4.5.1 Completing the Square Algebra Tile Investigation (continued) Standard Form Number of x 2 Tiles Number of x Tiles Number of Unit Tiles Sketch of the Square Length of the Square Area of the Square (Length) 2 Unit Tiles Left Over (+) Borrowed (-) Expression Combining Previous Two Columns x 2 + 2x x+1 (x+1) 2 2 (x+1) x 2 + 4x + 1 x 2 + 6x

23 4.5.1 Completing the Square Algebra Tile Investigation (continued) Standard Form Number of x 2 Tiles Number of x Tiles Number of Unit Tiles Sketch of The Square Length of the Square Area of the Square (Length) 2 Unit Tiles Left Over (+) Borrowed (-) Expression Combining Previous Two Columns 2x 2 + 4x + 5 3x x x 2 + 8x

24 Unit 4 : Day 6 : Completing the square using algebra Grade 11 U/C Minds On: 15 Action: 40 Consolidate:20 Total=75 min Minds On Action! Description/Learning Goals Express the equation of a quadratic function in the vertex form f(x) = a(x h) 2 + k, given the standard from f(x) = ax 2 + bx + c by completing the square, including cases where a b is a simple rational number, and verify, using graphing technology, that these forms are equivalent representations. Students will be able to abstractly change from the standard from of a quadratic function to the vertex form of the quadratic function. Whole Class Discussion Lead a discussion where the characteristics of quadratic functions that can and cannot use algebra tiles to complete the square are listed. Organize the characteristics using a mind map. Pairs Investigation In pairs, students will complete questions 1-4 of BLM using results from BLM and graphing calculators. Whole Class Discussion/Guided Instruction Have students share their procedures for completing the square. Consolidate their ideas into an agreed class procedure. Proposed guiding questions; o What is the same and what is different in the two forms of the equation? o What is the relationship between h and b? Materials BLM Graphing calculators FRAME document Assessment Opportunities For characteristics see lesson 5. Literacy Strategy Mind Mapping (Think Literacy: Cross-curricular Strategies Grades 10-12, p 32) Pairs Practice Students practice completing the square using the class procedure by completing question 5 on BLM Partners will correct each other s work. Consolidate Debrief Individual Create Graphic Organizer Students will summarize the procedure for completing the square algebraically by using a graphic organizer such as a flowchart. Mathematical Process Focus: Representing (In their journals, students will represent the procedure for completing the square using a graphic organizer) Home Activity or Further Classroom Consolidation Concept Practice Reflection Assign extra practice questions as needed. Update FRAME graphic organizer document with information from the last two lessons. 24

25 4.6.1 Completing the Square Algebraic Procedure Instructions: 1. Copy the vertex form of the expressions from BLM into column two below. 2. Using a graphing calculator, graph the expression in standard form and vertex form on the same graph. What do you notice about the two graphs? Record your observations in the Observations/Findings column of the table. 3. With your partner, propose a procedure for changing a quadratic function in standard form into the vertex form based on your findings. Record your response in the Observations/Findings column of the table. 4. Discuss your observations with your partner. Your teacher will ask you to share your ideas with the class. 5. Once your class agrees on a procedure, fill in the last column of the table by completing the square using the procedure the class agrees on. Standard Form Vertex Form Observations/ Findings Completing the Square Algebraically x 2 + 4x + 1 x 2 + 6x + 8 2x 2 + 4x + 5 3x x x 2 + 8x

26 Unit 4 : Day 9: CAS ing out the quadratic formula Grade 11 U/C Minds On: 15 Action: 40 Consolidate:20 Total=75 min Minds On Description/Learning Goals Materials To review solving simple quadratic equations BLM Explore the algebraic development of the quadratic formula with technology BLM Use various methods to solve quadratic equations BLM BLM Scissors Glue or tape CAS enabled calculator Assessment Opportunities Pairs Activity (Review of Solving Quadratic Equations) Students complete BLM4.9.1 using a Think/Pair/Share strategy. Action! Consolidate Debrief Pairs Activity (Putting the Pieces in Order) Students are presented with a quadratic equation that cannot be solved by factoring or other known methods. Thus a new method is introduced. Students apply their prior knowledge of completing the square to this new situation by correctly sequencing the steps that solve this type of quadratic equation. Refer to BLM4.9.2 Whole Class Demonstration ( CAS ing Out the Quadratic Formula) Teacher will first solve a numerical example of a quadratic equation using the CAS enabled calculator. Teacher will then connect the steps of the numeric example to the development of the quadratic formula. Students are only expected to follow the demonstration using BLM Mathematical Process Focus: Reasoning and Proving (Students will reason by correctly sequencing the steps in developing the quadratic formula.) Small Groups Consolidate Using a Round Table cooperative learning literacy strategy (refer to BLM4.9.4), students will practice solving quadratic equations using various methods. Students are not required to be able to develop the quadratic formula. Literacy Strategy: During Minds On use the Think, Pair, Share strategy. During Consolidate use the Round Table strategy. (Think Literacy: Cross-curricular Strategies: Gr 7-9, p 96) Home Activity or Further Classroom Consolidation Concept Practice Assign extra practice questions as needed. 26

27 4.9.1 Review of Solving Quadratic Equations Working with a partner, solve and check the equations given. If Partner A has solved the equation, partner B will check it and vice-versa, until all the equations have been solved and checked. PARTNER A Solve: x 5 = 0 PARTNER B Check: Check: Solve: x 2 = 25 Solve: x = 25 Check: Check: Solve: (x 2) 2 = 25 Solve: (x 2) = 25 Check: Check: Solve: 4(x 2) = 25 27

28 4.9.2 Putting the Pieces in Order All of the quadratic equations from the previous activity could be solved by isolating the variable. Sometimes it is not possible to solve by isolating the variable. Another method is required. We must perform the following steps: Group the variable terms on one side and constant terms on the other side of the equation Complete the square Factor into a perfect square Solve for x Example: The equation x 2 10x 3 = 0 has been solved by using the method of completing the square. However, the steps are not in the correct order. Working with a partner, cut out the steps given and rearrange them in the correct order. Glue these pieces in the space provided on the next page. x x = (x + 5) = ± 28 x = 0.29 or x = x x 3 = 0 (x + 5) 2 = 28 x x = x x = 3 x x = x x + (5) 2 (5) 2 = 3 x = -5 ±

29 4.9.2 Putting the Pieces in Order (continued) GLUE YOUR STEPS HERE: Using completing the square to solve a quadratic equation: Note: How would one check the solutions to this equation? 29

30 4.9.3 CAS ing Out the Quadratic Formula As you may have noticed, applying the method of completing the square will solve any quadratic equation in the form of ax 2 + bx +c = 0, but it is a tedious and repetitive process. Luckily, because it is so repetitive, the process can be generalized into a formula and then this formula can be quickly and easily applied to solve any quadratic equation, no matter how complicated it may look. In a sense, the quadratic formula is the nutcracker for solving all quadratics!! In this activity you will follow the development of the quadratic formula as shown through the use of a very powerful calculator. This calculator allows the inputting of instructions common to algebra, such as factor and expand. This calculator is called a CAS (Computer Algebra System) enabled calculator. The model name of the calculator is either a TI-89 or TI-92. Part A: In this activity you will follow the use of a Computer Algebra System (CAS) to solve a quadratic equation by completing the square. Use the screen shots to guide your work. The screen shots are from the TI-92. Description of Process Guiding Screen Shots Try to factor the trinomial 2x 2 + 8x - 9 = 0 using your TI-89 or TI-92 by pressing F2 and selecting 2:factor( After the opening bracket, enter the equation you are trying to solve, followed by a closing bracket. Press ENTER Note: When entering an equation, be sure to enter * when multiplying a coefficient with a variable (ex. 8x should be entered as 8 * x ) In solving this quadratic equation by completing the square the first step is rewrite the equation so that the variable terms are on one side of the equation and the constant term on the other. Thus, eliminate the constant term, which is -9, from both sides of the equation by entering + 9 (without the quotation marks) on the calculator, then press ENTER. 30

31 4.9.3 CAS ing Out the Quadratic Formula (continued) Description of Process Guiding Screen Shots Next, divide both sides of the equation by the coefficient of the x 2 term. In this case, enter 2. Press ENTER Next, expand the equation to separate all the terms. To do this press F2 and selecting 3:expand(. Use the arrow up key and select the equation by pressing ENTER. Don t forget to close this equation with an end bracket ). Press ENTER. Next, is to write in a term that completes the square on the left side of the equation. To do this we add an amount that is half of the coefficient of the x-term, squared, to both sides of the equation. In this case, enter +(4/2)^2. Press ENTER Next, factor both sides of the equation by pressing F2 and selecting 2:factor( Use the arrow up key and select the equation by pressing ENTER. Don t forget to close this equation with an end bracket ). Press ENTER. 31

32 4.9.3 CAS ing Out the Quadratic Formula (continued) Description of Process Guiding Screen Shots Next, take the square root of both sides by press the (2 nd X), then selecting your equation by scrolling up and pressing ENTER. Close your brackets and press ENTER. Next, solve for x by pressing F2 then select 1:solve( by pressing ENTER. Use the arrow up key and select the equation by pressing ENTER. Then enter, x ). What are your solutions? x = OR x = 32

33 4.9.3 CAS ing Out the Quadratic Formula (continued) Part B: Now we are ready to develop the quadratic formula. We will repeat the steps used above except this time we will apply them against the general form of the quadratic equation: ax 2 + bx + c = 0 Specific Numeric Example General Form Example 33

34 4.9.3 CAS ing Out the Quadratic Formula (continued) Specific Numeric Example General Form Example 34

35 4.9.3 CAS ing Out the Quadratic Formula (continued) Specific Numeric Example General Form Example Full Screen Below Wow, you ve just developed the quadratic formula! The Quadratic Formula is: x = b ± b 2 4ac 2a This will solve the general quadratic equation in the form ax 2 + bx + c = 0 35

36 4.9.4 The Quadratic Round Table (Teacher) In this exercise, students will practice solving quadratic equations using a structure called Roundtable Organize students into teams of 4 with desks facing each other. One sheet of paper is used per question, per team. Write out a question that is to be solved on the board or overhead. The goal is to have each team solve the quadratic, one step at a time, by having each student take turns in writing a step and passing the paper on to the next member of the team. Incorrect responses should be noted and corrected by other team members. This constitutes one turn for that member. When the team is satisfied that the quadratic has been solved, they should raise their hands and the teacher will write their final answer on the board. Answers can then be compared as a class. Possible quadratics to be solved: 1. 0 =4x 2 3x x 2 = -3x (x 5) 2 = x 2 6x = x 2 5x +2 =0 6. (x 4)(x+2)=0 Notes: 1. Some of these quadratics are solved more efficiently without the use of the quadratic formula. One can use this fact as a quick assessment of whether students are able to select different tools when solving quadratics. 2. Whether to ask students to state exact or approximate solutions to a desired number of decimal places is left open to the teacher. This may be a good point for a class-wide discussion. 3. This may or may not be done in a race format, depending on the group dynamics of the cass. 36

37 Unit 4 : Day 12 : The Nano Project or Fuel Fit Activities Grade 11 U/C Minds On: 15 Action: 40 Consolidate:20 Description/Learning Goals Collect data from a primary or a secondary source that can be modelled as a quadratic function without the use of technology. Display the data in tabular, graphical and algebraic forms without the use of technology Materials BLM BLM Total=75 min Minds On Whole Class Discussion (Nano Project or Fuel Fit) Teacher sets the context for either activity by posing either of the guiding questions below (refer to BLM or BLM ) Nano Project-If you are selling a product and want to maximize your income what factors do you have to consider? Assessment Opportunities There are two investigations to choose from: The Nano Project (BLM4.12.1) uses a primary data source or Action! Consolidate Debrief OR Fuel Fit-What factors influence fuel economy while driving? Small Groups Investigation (Nano Project or Fuel Fit) Teacher conducts survey as described in BLM then students will complete the table and investigate the relation between price and income. OR Students will investigate the relation between fuel consumption and speed (BLM4.12.2). Mathematical Process Focus: Connecting (Students will connect realworld data to mathematics. Students will represent the data in tabular, graphical and algebraic forms.) Whole Class Discussion Teacher could pose the following questions: a) How would your graph change if fewer data values were used? b) If student council had to pay a flat fee of $500 to purchase the Nanos, how would the graphical model be affected? Fuel Fit (BLM4.12.2) uses a secondary data source. Ask: Which model, numerical or graphical, is easiest to use when looking for trends in data? Reference for Fuel Fit: du/theses/available/ etd /unrestricted/ ETD.pdf ta/tedb25/spreadsh eets/table4_22.xls Application Concept Practice Reflection Home Activity or Further Classroom Consolidation Graph fuel consumption versus speed for cars in Determine an algebraic equation that models this curve. What speed offers optimal fuel consumption 37

38 The Nano Project The student council of your school is going to sell 4GB ipod Nano MP3 players to support the purchase of a school van. The council is interested in determining what a student would pay for each Nano to maximize their income. Your class has been selected as the sample for this study The survey of your class will be conducted in this manner: Your instructor will call out a dollar figure starting at $0 and increasing in increments of $10 up to $300. For each amount that you would be willing to pay for the Nano, raise your hand. You may raise your hand for more than one price until you have reached your limit. After each increment, your instructor will count the votes and you will record this figure in the table below under the column titled Frequency. You must calculate the Income column by multiplying the Frequency by the Purchase price. Since pictures are worth a thousand words, you will determine a graphical model of the data manually. This will be a scatter plot of Income vs. Purchase Price. Finally, you will determine an algebraic model that describes the trend of the data. ipod Purchase Survey (Raw Data) Purchase Price per ipod ($) Frequency Income ($) Purchase Price per ipod ($) Frequency Income ($) 38

39 The Nano Project (continued) ipod Purchase Survey (Scatter Plot) 39

40 The Nano Project (continued) Determining the Best Purchase Price You will now determine what purchase price will maximize income. Collect Coordinates to create An Algebraic Model: Draw a curve that best fits the trend of your data and identify the type of function. Record the coordinates of at least 5 points on the curve. Be sure to include the coordinates of the intercepts and the point where income is maximized. Record your data points in the table below: Purchase Price Per Nano ($) Income ($) Create three algebraic models: Factored Form Vertex Form Standard/Expanded Form y = a(x r)(x s) y = a(x h) 2 + k y = ax 2 + bx + c Solve for a by substituting the x-intercepts for r and s and the value of any other point for x and y. Solve for a by substituting the vertex for h and k and the value of any other point for x and y. Expand either the factored or vertex form to rewrite the equation in standard/expanded form. So, the equation of the quadratic in factored form is: So, the equation of the quadratic in vertex form is: So, the equation of the quadratic in standard/expanded form is: 40

41 The Nano Project (continued) Analysis: The student council requires the following information. Determine which quadratic form is most suitable to find the requested information. Place a check mark in this box. Then provide a calculation that provides an answer for the requested information Requested Information At which price(s) is/are the income from the Nano sales is $1000 At which price(s) is/are the income from the Nano sales zero If the purchase price of the Nano is $105, predict what would the income be At which price is the income from the sales maximized Factored Form Vertex Form Standard/Expanded Form 1. Compare your algebraic models with that of another group. Are they the same? Give reasons for any differences. 2. Explain why the graph would rise and then fall (increase then decrease)? 41

42 The Nano Project (Teacher) Sample Data Purchase Price per Nano ($) Frequency Income ($) Chart Title Income ($ Purchase Price Per Nano ($) y = x x

43 Fuel Fit (Teacher) In the near future you may be interested in getting your driver s license. With a partner answer these questions. Be prepared to share your answer. Jot down ways of reducing fuel consumption. If you are concerned about gas consumption, is it best to have a high value (maximum) for km/l or a low value (minimum) km/l. Explain your reasoning. 43

44 Fuel Fit (continued) The following data relates speed to fuel consumption based on an average of 9 different vehicles ( Speed (km/h) Fuel Consumption, Year 1973 (km/l) Fuel Consumption, Year 1984 (km/l) Fuel Consumption, Year 1997 (km/l) Make a scatter plot Gas Consumption vs. Speed for Draw the curve of best fit What type of function models the curve of best fit? 44

45 Fuel Fit (continued) Gas Consumption (Scatter Plot) 45

46 Fuel Fit (continued) Determining the Optimal Driving Speed You will now determine what speed you should drive a vehicle to maximize fuel consumption. Collect Coordinates to create An Algebraic Model: Draw a curve that best fits the trend of your data and identify the type of function. Collect the coordinate of at least 5 points on the curve. Be sure to include the coordinates of the intercepts, the high point (vertex). Record your data points in the table below: Speed (km/h) Fuel Consumption (km/l) Create three algebraic models: Factored Form Vertex Form Standard/Expanded Form y = a(x r)(x s) y = a(x h) 2 + k y = ax 2 + bx + c Solve for a by substituting the x-intercepts for r and s and the value of any other point for x and y. Solve for a by substituting the vertex for h and k and the value of any other point for x and y. Expand either the factored or vertex form to rewrite the equation in standard/expanded form. So, the equation of the quadratic in factored form is: So, the equation of the quadratic in vertex form is: So, the equation of the quadratic in standard/expanded form is: 46

47 Fuel Fit (continued) Analysis: The Ministry of Transportation wants you to make predictions using your algebraic models. Determine which quadratic form is most suitable in find the requested information. Place a check mark in this box. Then provide a calculation that provides an answer for the requested information Requested Information At which speed(s) is fuel consumption 12km/L Factored Form Vertex Form Standard/Expanded Form At which speed(s) is fuel consumption zero If the speed is 60 km/h, predict the expected fuel consumption Which speed gives the optimal fuel consumption 1. Compare your algebraic model with that of another group. Are they the same? Explain. 2. Explain why the graph would rise and then fall (increase then decrease)? 47

48 Fuel Fit (continued) 3. You are planning 500 km trip from Parabola Ville to Quadraborough. a. Using what you learned from this activity, sketch the relationship between Gas Consumption (km/l) and Speed (km/h). Speed (km/h) b. Predict the relationship between total cost in dollars of this trip and the speed at which this trip has been driven. Speed (km/h) 48

49 Unit 4 : Day 13 : The TI-Nano Project or Fathom Fuel Fit Activities Grade 11 U/C Minds On: 15 Action: 40 Consolidate:20 Total=75 min Minds On Action! Consolidate Debrief Reflection Description/Learning Goals Generate curves of best fit using technology Determine the equation of the curve using technology Compare different algebraic forms of quadratic equations Whole Class Four Corners Activity (BLM ) Teacher reads the context to the class and writes the first algebraic model on the board. Based on the algebraic model given, students decide on an answer and move to that corner. Teacher then posts the next algebraic model and students have the opportunity to revise their answers. Repeat for the last model. Note: Read the equations in the order given Individual Investigation (The TI-Nano Project or Fathom Fuel Fit) Students will use BLM to guide an investigation using technology on the TI-Nano Project OR Students will use BLM to guide an investigation using technology on the Fathom Fuel Fit Mathematical Process Focus: Representing (Students will represent applications of quadratics, graphically and algebraically using technology) Selecting Tools and Computational Strategies (Students will use appropriate technology to display and analyse quadratic models) Pairs Discussion Students compare their summary tables and revise if necessary. Whole Class Discussion Pose the following guiding questions: c) Which algebraic model of the quadratic function would you use to find the y-intercept? d) Which algebraic model of the quadratic function would you use to find the zeros (x-intercepts)? e) Which algebraic model of the quadratic function would you use to find the maximum or minimum value of the function? Home Activity or Further Classroom Consolidation In their journal, students write their response to: What are the benefits/drawbacks of using technology versus pencil and paper methods when determining quadratic models? Assessment Opportunities Materials BLM BLM BLM Graphing calculators FRAME document OR Computer lab There are two lessons for the teacher to choose. TheTI-83 Nano Project uses a primary data source and Fathom Fuel Fit uses a secondary data source. Teachers can use the file Fuel Fit Sliders.ftm which contains data and sliders. Literacy Strategy: During Minds On use the Four Corners strategy to review and consolidate the previous lesson. (Think Literacy: Cross-curricular Strategies Grades 10-12, p 106) Students can update their FRAME graphic organizer document for quadratic functions with new information acquired in the last two lessons. 49